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 A136424 a(n) = floor((x^n - (1-x)^n) / (2x-1) +.5) where x = (sqrt(6)+1)/2 (and hence 2x-1 = sqrt(6)). 0

%I #11 Jul 28 2021 06:59:02

%S 1,1,2,4,6,11,19,32,55,95,164,283,488,842,1451,2503,4318,7447,12844,

%T 22152,38207,65898,113657,196029,338101,583137,1005763,1734685,

%U 2991888,5160244,8900104,15350410,26475540,45663552,78757977,135837417

%N a(n) = floor((x^n - (1-x)^n) / (2x-1) +.5) where x = (sqrt(6)+1)/2 (and hence 2x-1 = sqrt(6)).

%C This is analogous to the closed form of the formula for the n-th Fibonacci number. Even before truncation, these numbers are rational and the decimal part always ends in 5. For x=(sqrt(6)+1)/2, a(n)/a(n-1) -> x.

%F The general form of x is (sqrt(r)+1)/2, r=1,2,3..

%F a(n) = floor(b(n)/2^n) where b(n) = 2*A002532(n)+2^(n-1). - _R. J. Mathar_, Sep 10 2016

%o (PARI) g(n,r) = for(m=1,n,print1(fib(m,r)",")) fib(n,r) = x=(sqrt(r)+1)/2;floor((x^n-(1-x)^n)/sqrt(r)+.5)

%K nonn

%O 1,3

%A _Cino Hilliard_, Apr 01 2008

%E Definition corrected by Frederic van der Plancke (fplancke(AT)hotmail.com), May 08 2009

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Last modified September 21 18:47 EDT 2023. Contains 365503 sequences. (Running on oeis4.)